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Spring Motion Calculator

Calculate spring force, period, frequency, and analyze simple harmonic motion in spring-mass systems

Parameters

kgⓘ
N/mⓘ
mⓘ
Show Trail

Controls

xⓘ

Calculated Values

Period:
1.99;s1.99;s
Frequency:
0.50;Hz0.50;Hz
Angular Frequency:
3.16;rad/s3.16;rad/s
Total Energy:
1.25;J1.25;J
Maximum Force:
5.00;N5.00;N
Maximum Velocity:
1.58;m/s1.58;m/s

Examples

Example 1: Light Spring

A 1 kg mass on a spring with k = 10 N/m and amplitude 0.5 m.

  • Period: 1.991.99
  • Frequency: 0.500.50
  • Angular Frequency: 3.163.16
  • Total Energy: 1.251.25

Example 2: Stiff Spring

A 2 kg mass on a stiff spring with k = 100 N/m and amplitude 0.2 m.

  • Period: 0.890.89
  • Frequency: 1.131.13
  • Angular Frequency: 7.077.07
  • Total Energy: 2.002.00

Example 3: Heavy Mass

A 5 kg mass on a spring with k = 20 N/m and amplitude 1.0 m.

  • Period: 3.143.14
  • Frequency: 0.320.32
  • Angular Frequency: 2.002.00
  • Total Energy: 10.0010.00

Visualization

Spring Motion and Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium: F = -kx, where k is the spring constant and x is the displacement. The negative sign indicates that the force is restorative (opposes the displacement).

When a mass is attached to a spring and displaced from equilibrium, the system undergoes simple harmonic motion. The restoring force provided by the spring creates oscillatory motion around the equilibrium position.

The period of oscillation for a spring-mass system is given by T = 2π√(m/k), where m is the mass and k is the spring constant. This period is independent of the amplitude of oscillation.

The frequency of oscillation is the reciprocal of the period: f = 1/T = (1/2π)√(k/m). The angular frequency is ω = √(k/m).

The total mechanical energy of the system is conserved and equals the maximum potential energy: E = ½kA², where A is the amplitude of oscillation.

Key Concepts

  • Hooke's Law: F = -kx (restorative force)
  • Period: T = 2π√(m/k) (independent of amplitude)
  • Frequency: f = (1/2π)√(k/m)
  • Angular Frequency: ω = √(k/m)
  • Total Energy: E = ½kA² (conserved)
  • Simple Harmonic Motion: x(t) = A cos(ωt)

Real-World Applications

  • Automotive suspension systems
  • Musical instruments (strings, reeds)
  • Seismic isolation systems
  • Mechanical watches and clocks
  • Vibration dampers and shock absorbers
  • Atomic force microscopy

Explore Further

More mechanics tools

Physics Equations

Hooke's Law:
F=−kxF = -kx
Period:
T=2πmkT = 2\pi\sqrt{\frac{m}{k}}
Frequency:
f=12πkmf = \frac{1}{2\pi}\sqrt{\frac{k}{m}}
Angular Frequency:
ω=km\omega = \sqrt{\frac{k}{m}}
Total Energy:
E=12kA2E = \frac{1}{2}kA^2
Displacement:
x(t)=Acos⁡(ωt)x(t) = A\cos(\omega t)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Angular Frequency

First, calculate the angular frequency using the formula:

Equation:

ω=km\omega = \sqrt{\frac{k}{m}}

Calculation:

ω=10.01.0=3.16 rad/s\omega = \sqrt{\frac{10.0}{1.0}} = 3.16 \text{ rad/s}

Explanation:

The angular frequency determines how fast the spring oscillates.

2

Step 2: Calculate Period

The period is the time for one complete oscillation:

Equation:

T=2πω=2πmkT = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}

Calculation:

T=2π3.16=1.99 sT = \frac{2\pi}{3.16} = 1.99 \text{ s}

Explanation:

The period is independent of the amplitude of oscillation.

3

Step 3: Calculate Frequency

Frequency is the reciprocal of the period:

Equation:

f=1Tf = \frac{1}{T}

Calculation:

f=11.99=0.50 Hzf = \frac{1}{1.99} = 0.50 \text{ Hz}

Explanation:

Frequency tells us how many oscillations occur per second.

4

Step 4: Calculate Total Energy

The total mechanical energy is conserved and equals the maximum potential energy:

Equation:

E=12kA2E = \frac{1}{2}kA^2

Calculation:

E=12(10.0)(0.5)2=1.25 JE = \frac{1}{2}(10.0)(0.5)^2 = 1.25 \text{ J}

Explanation:

This energy oscillates between kinetic and potential forms during the motion.

Frequently Asked Questions (FAQ)

What is Hooke's Law?

Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium: F = -kx. The negative sign indicates that the force is restorative and opposes the displacement.

Why is the period independent of amplitude?

The period depends only on the mass and spring constant, not the amplitude. This is a characteristic of simple harmonic motion where the restoring force is proportional to displacement.

What happens to the period if the mass is doubled?

The period increases by a factor of √2 (approximately 1.414) when the mass is doubled, since T ∝ √m.

What happens to the frequency if the spring constant is quadrupled?

The frequency doubles when the spring constant is quadrupled, since f ∝ √k.

Is energy conserved in spring motion?

Yes, the total mechanical energy (kinetic + potential) is conserved in ideal spring motion. The energy oscillates between kinetic and potential forms.

Practice MCQs

  1. According to Hooke's Law, the spring force is:
  2. The period of a spring-mass system depends on:
  3. If the spring constant is doubled, the frequency becomes:
  4. The total energy of a spring-mass system is:
  5. In simple harmonic motion, the acceleration is:

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